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Properties

Label 1-23e2-529.346-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.506 - 0.862i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.460 + 0.887i)2-s + (0.460 + 0.887i)3-s + (−0.576 + 0.816i)4-s + (−0.0682 + 0.997i)5-s + (−0.576 + 0.816i)6-s + (−0.576 − 0.816i)7-s + (−0.990 − 0.136i)8-s + (−0.576 + 0.816i)9-s + (−0.917 + 0.398i)10-s + (0.682 + 0.730i)11-s + (−0.990 − 0.136i)12-s + (−0.917 − 0.398i)13-s + (0.460 − 0.887i)14-s + (−0.917 + 0.398i)15-s + (−0.334 − 0.942i)16-s + (−0.917 − 0.398i)17-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)2-s + (0.460 + 0.887i)3-s + (−0.576 + 0.816i)4-s + (−0.0682 + 0.997i)5-s + (−0.576 + 0.816i)6-s + (−0.576 − 0.816i)7-s + (−0.990 − 0.136i)8-s + (−0.576 + 0.816i)9-s + (−0.917 + 0.398i)10-s + (0.682 + 0.730i)11-s + (−0.990 − 0.136i)12-s + (−0.917 − 0.398i)13-s + (0.460 − 0.887i)14-s + (−0.917 + 0.398i)15-s + (−0.334 − 0.942i)16-s + (−0.917 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (346, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5517263915 + 0.9644160091i\)
\(L(\frac12)\) \(\approx\) \(-0.5517263915 + 0.9644160091i\)
\(L(1)\) \(\approx\) \(0.5259554157 + 0.9727469642i\)
\(L(1)\) \(\approx\) \(0.5259554157 + 0.9727469642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.460 + 0.887i)T \)
3 \( 1 + (0.460 + 0.887i)T \)
5 \( 1 + (-0.0682 + 0.997i)T \)
7 \( 1 + (-0.576 - 0.816i)T \)
11 \( 1 + (0.682 + 0.730i)T \)
13 \( 1 + (-0.917 - 0.398i)T \)
17 \( 1 + (-0.917 - 0.398i)T \)
19 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (0.682 + 0.730i)T \)
31 \( 1 + (-0.775 - 0.631i)T \)
37 \( 1 + (-0.334 + 0.942i)T \)
41 \( 1 + (0.203 - 0.979i)T \)
43 \( 1 + (0.962 + 0.269i)T \)
47 \( 1 + (-0.775 + 0.631i)T \)
53 \( 1 + (-0.917 - 0.398i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (-0.0682 + 0.997i)T \)
67 \( 1 + (0.682 + 0.730i)T \)
71 \( 1 + (0.203 + 0.979i)T \)
73 \( 1 + (0.682 - 0.730i)T \)
79 \( 1 + (0.962 + 0.269i)T \)
83 \( 1 + (-0.0682 + 0.997i)T \)
89 \( 1 + (0.854 - 0.519i)T \)
97 \( 1 + (0.203 + 0.979i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.87899491758106194736918689251, −21.87003737422590139509511343241, −21.360115100934685274989782784340, −20.108971657266177441694099739711, −19.58341889939986676090944411278, −19.19628436169561111744319381079, −18.05404899028529243631614861649, −17.27494509278168682334577761139, −15.991312539378274018010914338282, −15.01049637765440779268145954602, −14.03055434145671495072734714182, −13.29712462353191070079481198289, −12.52861546036253051367712864646, −11.99950201531194225991600596743, −11.16604905224701754282920211807, −9.43163499699942545467842160320, −9.14978264004937215314008509150, −8.27189204928063665828409245007, −6.74128463878463189954438231255, −5.88068774484919227295222076514, −4.83788259342531330863196271330, −3.66651428623097587621196120192, −2.60117257256995383664201444995, −1.74028970827608557104391729932, −0.45432004404128421911886773947, 2.508548273835544484266342125797, 3.48407382524028681506457349388, 4.16668312796238955243556339794, 5.165797733597219019709686627455, 6.45869487191352330648091916930, 7.199877245290957173786345584046, 7.97610317326597234277567060198, 9.36020873587287173447774105021, 9.89258129006983692031435702888, 10.91423714889765598596869336584, 12.11079918678668404992132576207, 13.23487268148296789531538934495, 14.2727053503350113085227195076, 14.52818156412621560309937864697, 15.50104117033844316698168846291, 16.168879423438521615736113700711, 17.14117694856177855479450994649, 17.78541887083201870802521246916, 19.08061954641907103236325918900, 19.98360333501203879044526355043, 20.73106197250271099076833228681, 22.01561986258716173058744202238, 22.456279062453304573376252966486, 22.82490300245156768004691615830, 24.03088477176395118690297609453

Graph of the $Z$-function along the critical line